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Preconditioned Krylov subspace and GMRHSS iteration methods for solving the nonsymmetric saddle point problems

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Abstract

In the present paper, we propose a separate approach as a new strategy to solve the saddle point problem arising from the stochastic Galerkin finite element discretization of Stokes problems. The preconditioner is obtained by replacing the (1,1) and (1,2) blocks in the RHSS preconditioner by others well chosen and the parameter α in (2,2) −block of the RHSS preconditioner by another parameter β. The proposed preconditioner can be used as a preconditioner corresponding to the stationary itearative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergence properties of the GMRHSS iteration method are derived. Meanwhile, we analyzed the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Finally, numerical results show the effectiveness of the proposed preconditioner as compared with other preconditioners.

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Acknowledgments

The authors would like to thank Laura Dykes for enlightening comments and corrections on an early draft of this manuscript and would like to express their sincere thanks to the referees for their most valuable suggestions.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées et Informatique, Faculté des Sciences et Techniques- Gueliz, Marrakech, Morocco

    A. Badahmane & A. H. Bentbib

  2. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral Côte d’Opale, Batiment H. Poincarré, 50 rue F. Buisson, F-62280, Calais Cedex, France

    A. Badahmane & H. Sadok

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  1. A. Badahmane
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  3. H. Sadok
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Correspondence to H. Sadok.

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Badahmane, A., Bentbib, A.H. & Sadok, H. Preconditioned Krylov subspace and GMRHSS iteration methods for solving the nonsymmetric saddle point problems. Numer Algor 84, 1295–1312 (2020). https://doi.org/10.1007/s11075-019-00833-4

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